This concerns some of the discussion by Franzen, Simpson and Shipman that
my paper "Does mathematics need new axioms?" has sparked, namely
concerning my conjecture that all or almost all of scientifically
applicable mathematics is formalizable in a certain system W which is a
conservative extension of PA. (The designation 'W' is in honor of Hermann
Weyl, and embodies a modern extension of his program in the 1918
monograph, _Das Kontinuum_. W was first described in a paper of mine
called "Weyl vindicated", not easily accessible from its Italian
publication but to re-appear in my forthcoming book of essays. The proof
that W is conservative over PA
is contained in my paper with G. Jaeger, "Systems of explicit mathematics
with non-constructive mu-operator II", Annals of Pure and Applied Logic
79 (1996), 37-52.)
First, some points of clarification are necessary.
1. It is not stated that physics, for example, can be done directly in PA
in any reasonable sense of the word. W is a theory which allows one to
talk about various kinds of sets and functions satisfying closure
conditions that permit one to define the real numbers, function spaces of
various kinds, and to develop substantial portions of modern functional
analysis more or less directly. As Simpson points out, through the work
on reverse mathematics (mainly by him and his students) much of this can
already be done in systems conservative over Primitive Recursive
Arithmetic, PRA. (A side issue is that the reverse math people work in
second order systems; this makes it easier to establish reverse
implications. W is a flexible finite type system (types are variable); it
is more amenable to direct formalization of practice, to the extent that
can be done at all in W.)
2. The verification of my conjecture comes from working through a body
of material in standard functional analysis texts through spectral theory
of bounded self-adjoint operators on a separable Hilbert space. All this
is in unpublished notes, but fairly carefully worked up. I have
done some preliminary work on extending this to unbounded self-adjoint
operators on such spaces, but don't have this written up, even in notes.
3. Is that the sum total of scientifically applicable mathematics? Well,
of course, who knows what the future will demand. So we're only talking
about what we observe to date. And in that respect I can only rely on
what experts tell me and what challenges are raised to the conjecture.
Two such challenges were brought to my attention. The first concerns the
use of non-separable spaces in a book by Emch (1972) to deal with systems
with an infinite number of degrees of freedom in quantum mechanics and
statistical mechanics. The second concerns the appearance of
non-measurable sets in a proposed hidden variable theory by Pitowsky
(1989) for quantum mechanics.
4. These raise prima-facie difficulties for my conjecture because:
(a) W does not prove the existence of non-(Lebesgue)-measurable sets of
reals.
(b) There is no obvious way to deal with analysis on non-separable spaces
in W.
Here, (a) is established by a suitable model of W, not by a reversal in
Simpson's sense. (b) is nothing I can prove; we just haven't seen how to
do it.
5. Both Emch's and Pitowsky's proposals are controversial theoretical
models. For example, the need for non-separable spaces is disputed by
Streater and Wightman (1978) p.87. And in his critical notice of
Pitowsky's work, Malament in Phil.of Science (1992) has argued that P's
proposal doesn't make sense and doesn't work. Among other things, P's
unusual probability measures are not even finitely additive. Moreover, it
did not seem that P. himself was committed to his proposal.
6. On these and related matters, I have consulted my distinguished
colleagues in applied mathematics, Joe Keller and George Papanicolao.
A communication from the latter concluded with: "I cannot think of a
compelling physical situation where non-measurability or non-separability
are essential."
7. Other challenges besides these kind might be raised. Shipman has
referred to two papers containing such. I have only just seen these, and
so cannot comment. His 1990 paper in TAMS 321 seems to suggest the need
for large cardinals (e.g. real-valued measurable) in some physical
applications, in models due to Pitowsky [sic!] and Gudder. I'm in no
position to evaluate these models, but as you can imagine, I am extremely
skeptical. What I'm going by is the bread and butter analysis that is
used in everyday mathematical physics.
8. Strong concepts and assumptions may have both practical (cf. Franzen's
remarks), heuristic and other kinds of instrumental values. I don't deny
this, in fact agree. But that's a larger issue--also touching on the
possible value of large cardinal axioms in combinatorics and
elsewhere--that I plan to address separately.
Category theory will have to wait.